Some New Fixed Point Theorems in Partial Metric Spaces with

Hindawi Journal of Function Spaces Volume 2017, Article ID 1072750, 13 pages https://doi.org/10.1155/2017/1072750

Research Article Some New Fixed Point Theorems in Partial Metric Spaces with Applications Rajendra Pant,1 Rahul Shukla,1 H. K. Nashine,2 and R. Panicker3 1

Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India Department of Mathematics, Amity University, Chhattisgarh Manth/Kharora (Opp. ITBP) SH 9, Raipur, Chhattisgarh 493225, India 3 Department of Mathematical Sciences and Computing, Walter Sisulu University, Mthatha 5117, South Africa 2

Correspondence should be addressed to R. Panicker; [email protected] Received 2 February 2017; Accepted 7 March 2017; Published 29 March 2017 Academic Editor: Tomonari Suzuki Copyright © 2017 Rajendra Pant et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, a number of fixed point theorems for contraction type mappings in partial metric spaces have been obtained by various authors. Most of these theorems can be obtained from the corresponding results in metric spaces. The purpose of this paper is to present certain fixed point results for single and multivalued mappings in partial metric spaces which cannot be obtained from the corresponding results in metric spaces. Besides discussing some useful examples, an application to Volterra type system of integral equations is also discussed.

Dedicated to Professor Shyam Lal Singh on his 75th birthday

1. Introduction and Preliminaries Throughout this paper N, R, and R+ denote the set of all natural numbers, the set of all real numbers, and the set of all nonnegative real numbers, respectively. The well-known Banach contraction theorem (BCT) has been generalized and extended by many authors in various ´ c [1] introduced the notion of quasiways. In 1974, Ciri´ contraction and obtained a forceful generalization of Banach contraction theorem. Definition 1. A self-mapping 𝑇 of a metric space 𝑋 is a quasicontraction if there exists a number 𝑟 ∈ [0, 1) such that, for all 𝑥, 𝑦 ∈ 𝑋, 𝑑 (𝑇𝑥, 𝑇𝑦) ≤ 𝑟𝑚 (𝑥, 𝑦) ,

(C)

where 𝑚(𝑥, 𝑦) = max{𝑑(𝑥, 𝑦), 𝑑(𝑥, 𝑇𝑥), 𝑑(𝑦, 𝑇𝑦), 𝑑(𝑥, 𝑇𝑦), 𝑑(𝑦, 𝑇𝑥)}. Theorem 2 (see [1]). A quasi-contraction on a complete metric space has a unique fixed point.

We remark that a quasi-contraction for a self-mapping on a metric space is considered as the most general among contractions listed by Rhoades [2]. In 2006, Proinov [3] established an equivalence between two types of generalizations of the BCT. The first type involves Meir-Keeler [4] type contraction conditions and the second type involves Boyd and Wong [5] type contraction conditions. Further, generalizing certain results of Jachymski [6] and Matkowski [7] he obtained the following ´ c’s quasigeneral fixed point theorem, which extends Ciri´ contraction. Theorem 3 (see [3], Th. 4.1). Let 𝑇 be a continuous and asymptotically regular self-mapping on a complete metric space (𝑋, 𝑑) satisfying the following conditions: (P1) 𝑑(𝑇𝑥, 𝑇𝑦) ≤ 𝜑(𝐷(𝑥, 𝑦)) for all 𝑥, 𝑦 ∈ 𝑋, (P2) 𝑑(𝑇𝑥, 𝑇𝑦) < 𝐷(𝑥, 𝑦), whenever 𝐷(𝑥, 𝑦) ≠ 0,

2

Journal of Function Spaces

where 𝜑 : R+ → R+ , a function satisfying the following: for any 𝜀 > 0 there exists 𝛿 > 𝜀 such that 𝜀 < 𝑡 < 𝛿 implies 𝜑(𝑡) ≤ 𝜀, 𝛾 ≥ 0, and 𝐷 (𝑥, 𝑦) = 𝑑 (𝑥, 𝑦) + 𝛾 [𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)] .

(1)

Then 𝑇 has a unique fixed point. Moreover if 𝛾 = 1 and 𝜑 is continuous with 𝜑(𝑡) < 𝑡 for all 𝑡 > 0, then the continuity of 𝑇 can be dropped. A mapping 𝑇 satisfying the conditions (P1) and (P2) is called a Proinov contraction. The following example shows the generality of Proinov contraction over quasi-contraction. Example 4 (see [8]). Let 𝑋 = {1, 2, 3} with the usual metric 𝑑 and 𝑇 : 𝑋 → 𝑋 such that 𝑇1 = 1,

(p3 ) 𝑝(𝑥, 𝑦) = 𝑝(𝑦, 𝑥); (p4 ) 𝑝(𝑥, 𝑧) ≤ 𝑝(𝑥, 𝑦) + 𝑝(𝑦, 𝑧) − 𝑝(𝑦, 𝑦). The pair (𝑋, 𝑝) is called a partial metric space. A partial metric 𝑝 on 𝑋 generates a 𝑇0 -topology 𝜏𝑝 on 𝑋 with a base of the family of open 𝑝-balls {𝐵𝑝 (𝑥, 𝑟) : 𝑥 ∈ 𝑋, 𝑟 > 0}, where 𝐵𝑝 (𝑥, 𝑟) = {𝑦 ∈ 𝑋 : 𝑝 (𝑥, 𝑦) < 𝑝 (𝑥, 𝑥) + 𝑟} .

(3)

If 𝑝 is a partial metric on 𝑋, then the function 𝑝𝑠 : 𝑋 × 𝑋 → R+ given by 𝑝𝑠 (𝑥, 𝑦) = 2𝑝 (𝑥, 𝑦) − 𝑝 (𝑥, 𝑥) − 𝑝 (𝑦, 𝑦)

(4)

for all 𝑥, 𝑦 ∈ 𝑋 is a metric on 𝑋.

(2)

𝑇3 = 1.

Example 6 (see [10, 14]). Let 𝑋 = R+ and 𝑝 : 𝑋 × 𝑋 → R+ given by 𝑝(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ 𝑋. Then (𝑋, 𝑝) is a partial metric space.

The mapping 𝑇 does not satisfy the condition (C). However, 𝑇 satisfies the conditions (P1) and (P2) with 𝜑(𝑡) = 2𝑡/(1 + 𝛾), where 𝛾 > 1.

Example 7 (see [9, 14]). Let 𝑋 = {[𝑎, 𝑏] : 𝑎, 𝑏 ∈ R, 𝑎 ≤ 𝑏}. Then 𝑝([𝑎, 𝑏], [𝑐, 𝑑]) = max{𝑏, 𝑑} − min{𝑎, 𝑐} defines a partial metric 𝑝 on 𝑋.

On the other hand, in 1994, Matthews [9] introduced the notion of partial metric spaces to study the denotational semantics of dataflow networks. It is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation (see [10] and references thereof). Matthews also obtained the partial metric version of Banach contraction theorem. Subsequently, many authors studied partial metric spaces and their topological properties and obtained a number of fixed point theorems for single and multivalued mappings (cf. [9–27] and many others). In [28], Haghi et al. pointed out that some fixed point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces. To demonstrate facts they considered certain cases. Motivated by Proinov’s results, in this paper, we present some fixed point theorems in partial metric spaces which cannot be obtained from the corresponding results in metric spaces. Indeed, we obtain some fixed and common fixed point theorems for single and multivalued mappings in the setting of partial metric spaces. Our results complement, extend, and generalize a number of fixed point theorems including some recent results in [10, 11, 14, 16, 23] and others. Besides discussing some useful examples, an application to Volterra type system of integral equations is also given. Finally, we show that fixed point problems discussed herein are well-posed and have limit shadowing property. For the sake of completeness, we recall the following definitions and results from [9, 10, 14].

Definition 8. Let (𝑋, 𝑝) be a partial metric space. Then one has the following:

𝑇2 = 3,

Definition 5. A partial metric on a nonempty set 𝑋 is a function 𝑝 : 𝑋 × 𝑋 → R+ such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋 (p1 ) 𝑝(𝑥, 𝑥) = 𝑝(𝑦, 𝑦) = 𝑝(𝑥, 𝑦) if and only if 𝑥 = 𝑦; (p2 ) 𝑝(𝑥, 𝑥) ≤ 𝑝(𝑥, 𝑦);

(1) A sequence {𝑥𝑛 } in 𝑋 converges to a point 𝑥 ∈ 𝑋 if and only if lim𝑛→∞ 𝑝(𝑥, 𝑥𝑛 ) = 𝑝(𝑥, 𝑥). (2) A sequence {𝑥𝑛 } in 𝑋 is Cauchy if lim𝑛,𝑚→∞ 𝑝(𝑥𝑛 , 𝑥𝑚 ) exists and is finite. (3) 𝑋 is complete if every Cauchy sequence {𝑥𝑛 } in 𝑋 converges to a point 𝑥 ∈ 𝑋, that is, 𝑝(𝑥, 𝑥) = lim𝑛,𝑚→∞ 𝑝(𝑥𝑛 , 𝑥𝑚 ). Lemma 9 (see [9]). Let (𝑋, 𝑝) be a partial metric space. Then 𝑋 is complete if and only if the metric space (𝑋, 𝑝𝑠 ) is complete. Furthermore, lim𝑛→∞ 𝑝𝑠 (𝑥𝑛 , 𝑧) = 0 if and only if 𝑝 (𝑧, 𝑧) = lim 𝑝 (𝑥𝑛 , 𝑧) = lim 𝑝 (𝑥𝑛 , 𝑥𝑚 ) . 𝑛→∞

𝑛,𝑚→∞

(5)

In [25], Romaguera introduced the following notions of 0-Cauchy sequence and 0-complete partial metric spaces. He obtained a characterization of completeness for partial metric space using the notion of 0-completeness. Definition 10. A sequence {𝑥𝑛 } in a partial metric space (𝑋, 𝑝) is 0-Cauchy if lim𝑛,𝑚→∞ 𝑝(𝑥𝑛 , 𝑥𝑚 ) = 0. The partial metric space (𝑋, 𝑝) is 0-complete if each 0-Cauchy sequence in 𝑋 converges to a point 𝑥 ∈ 𝑋 such that 𝑝(𝑥, 𝑥) = 0. Notice that every 0-Cauchy sequence in (𝑋, 𝑝) is Cauchy in (𝑋, 𝑝𝑠 ) and every complete partial metric space is 0complete. However, a 0-complete partial metric space need not be complete (cf. [29] and [25]). A subset 𝐴 of 𝑋 is closed (resp., compact) in (𝑋, 𝑝) if it is closed (resp., compact) with respect to the topology 𝜏𝑝 induced by 𝑝 on 𝑋. The subset 𝐴 is bounded in (𝑋, 𝑝) if there


3

exist 𝑥0 ∈ 𝑋 and 𝑀 > 0 such that 𝑎 ∈ 𝐵𝑝 (𝑥0 , 𝑀) for all 𝑎 ∈ 𝐴; that is, 𝑝 (𝑥0 , 𝑎) < 𝑝 (𝑎, 𝑎) + 𝑀 ∀𝑎 ∈ 𝐴.

Lemma 14 (see [28]). Let (𝑋, 𝑝) be a partial metric space, 𝑇 : 𝑋 → 𝑋 a self-mapping, 𝑑 the metric constructed in Proposition 13, and 𝑥, 𝑦 ∈ 𝑋. Define

(6) 𝑀𝑑 (𝑥, 𝑦) = max {𝑑 (𝑥, 𝑦) , 𝑑 (𝑥, 𝑇𝑥) , 𝑑 (𝑦, 𝑇𝑦) ,

𝑝

Let 𝐶𝐵 (𝑋) be the collection of all nonempty, closed, and bounded subsets of 𝑋 with respect to the partial metric 𝑝. For 𝐴 ∈ 𝐶𝐵𝑝 (𝑋), one defines 𝑝 (𝑥, 𝐴) = inf 𝑝 (𝑥, 𝑦) . 𝑦∈𝐴

1 [𝑑 (𝑥, 𝑇𝑦) + 𝑑 (𝑦, 𝑇𝑥)]} , 2

(9)

𝑀𝑝 (𝑥, 𝑦) = max {𝑝 (𝑥, 𝑦) , 𝑝 (𝑥, 𝑇𝑥) , 𝑝 (𝑦, 𝑇𝑦) ,

(7)

1 [𝑝 (𝑥, 𝑇𝑦) + 𝑝 (𝑦, 𝑇𝑥)]} . 2

For 𝐴, 𝐵 ∈ 𝐶𝐵𝑝 (𝑋), 𝛿𝑝 (𝐴, 𝐵) = sup𝑝 (𝑎, 𝐵) ,

Then 𝑀𝑑 (𝑥, 𝑦) = 𝑀𝑝 (𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋 with 𝑥 ≠ 𝑦.

𝛿𝑝 (𝐵, 𝐴) = sup𝑝 (𝑏, 𝐴) ,

Using Proposition 13 and Lemma 14 above, we obtain the following result.

𝑎∈𝐴

𝑏∈𝐵

(8)

𝐻𝑝 (𝐴, 𝐵) = max {𝛿𝑝 (𝐴, 𝐵) , 𝛿𝑝 (𝐵, 𝐴)} .

Lemma 15. Let (𝑋, 𝑝) be a partial metric space and 𝑇 : 𝑋 → 𝑋 a self-mapping. Suppose 𝑑 : 𝑋 × 𝑋 → R+ is the constructed metric in Proposition 13 and 𝑥, 𝑦 ∈ 𝑋. Define

Proposition 11 (see [14]). Let (𝑋, 𝑝) be a partial metric space. For any 𝐴, 𝐵, 𝐶 ∈ 𝐶𝐵𝑝 (𝑋), one has

𝜇𝑑 (𝑥, 𝑦) = 𝑑 (𝑥, 𝑦) + 𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦) , 𝜇𝑝 (𝑥, 𝑦) = 𝑝 (𝑥, 𝑦) + 𝑝 (𝑥, 𝑇𝑥) + 𝑝 (𝑦, 𝑇𝑦) .

(𝛿1 ) 𝛿𝑝 (𝐴, 𝐴) = sup{𝑝(𝑎, 𝑎) : 𝑎 ∈ 𝐴}; (𝛿2 ) 𝛿𝑝 (𝐴, 𝐴) ≤ 𝛿𝑝 (𝐴, 𝐵); (𝛿3 ) 𝛿𝑝 (𝐴, 𝐵) = 0 implies that 𝐴 ⊆ 𝐵; (𝛿4 ) 𝛿𝑝 (𝐴, 𝐵) ≤ 𝛿𝑝 (𝐴, 𝐶) + 𝛿𝑝 (𝐶, 𝐵) − inf 𝑐∈𝐶𝑝(𝑐, 𝑐). Proposition 12 (see [14]). Let (𝑋, 𝑝) be a partial metric space. For any 𝐴, 𝐵, 𝐶 ∈ 𝐶𝐵𝑝 (𝑋), one has (H1) 𝐻𝑝 (𝐴, 𝐴) ≤ 𝐻𝑝 (𝐴, 𝐵); (H2) 𝐻𝑝 (𝐴, 𝐵) = 𝐻𝑝 (𝐵, 𝐴);

Then (a) 𝜇𝑑 (𝑥, 𝑦) ≤ 𝜇𝑝 (𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋; (b) 𝑀𝑝 (𝑥, 𝑦) ≤ 𝜇𝑝 (𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋. Proof. To prove (a), we shall consider three cases and the rest of the cases will follow in the same manner. Case 1 (𝑥 = 𝑦). One has 𝜇𝑑 (𝑥, 𝑦) = 0 + 𝑑 (𝑥, 𝑇𝑥) + 𝑑 (𝑦, 𝑇𝑦)

(H3) 𝐻𝑝 (𝐴, 𝐵) ≤ 𝐻𝑝 (𝐴, 𝐶) + 𝐻𝑝 (𝐶, 𝐵) − inf 𝑐∈𝐶𝑝(𝑐, 𝑐);

≤ 𝑝 (𝑥, 𝑥) + 𝑝 (𝑥, 𝑇𝑥) + 𝑝 (𝑦, 𝑇𝑦)

(H4) 𝐻𝑝 (𝐴, 𝐵) = 0 ⇒ 𝐴 = 𝐵. But the converse is not true.

= 𝜇𝑝 (𝑥, 𝑦) .

In view of Propositions 11 and 12, 𝐻𝑝 is a partial Hausdorff metric induced by the partial metric 𝑝.

𝜇𝑑 (𝑥, 𝑦) = 𝑑 (𝑥, 𝑦) + 0 + 𝑑 (𝑦, 𝑇𝑦) ≤ 𝑝 (𝑥, 𝑦) + 𝑝 (𝑥, 𝑥) + 𝑝 (𝑦, 𝑇𝑦)

Hitzler and Seda [19] obtained the following result to establish a relation between a partial metric and the corresponding metric on a nonempty set 𝑋.

= 𝜇𝑝 (𝑥, 𝑦) .

The following lemma is the key result in [28].

(11)

Case 2 (𝑥 = 𝑇𝑥). One has

2. Auxiliary Results

Proposition 13 (see [19, 28]). Let (𝑋, 𝑝) be a partial metric space. Then the function 𝑑 : 𝑋 × 𝑋 → R+ defined by 𝑑(𝑥, 𝑦) = 0 whenever 𝑥 = 𝑦 and 𝑑(𝑥, 𝑦) = 𝑝(𝑥, 𝑦) whenever 𝑥 ≠ 𝑦 is a metric on 𝑋 such that 𝜏𝑝𝑠 ⊆ 𝜏𝑑 . Moreover, (𝑋, 𝑑) is complete if and only if (𝑋, 𝑝) is 0-complete.

(10)

(12)

Case 3 (𝑦 = 𝑇𝑦). One has 𝜇𝑑 (𝑥, 𝑦) = 𝑑 (𝑥, 𝑦) + 𝑑 (𝑥, 𝑇𝑥) + 0 ≤ 𝑝 (𝑥, 𝑦) + 𝑝 (𝑥, 𝑇𝑥) + 𝑝 (𝑦, 𝑦) = 𝜇𝑝 (𝑥, 𝑦) . The proof of (b) follows easily form [8, page 3300].

(13)

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3. Single Valued Mappings For the sake of brevity, in this section, we shall use the following denotations: +

(2) Ψ the class of functions 𝜓 : R+ → R+ such that 𝜓 is upper semicontinuous from the right satisfying 𝜓(𝑡) < 𝑡 for all 𝑡 > 0.

(4) 𝛼(𝑥, 𝑦) = 𝑝(𝑆𝑥, 𝑇𝑦) + 𝑝(𝐴𝑥, 𝑆𝑥) + 𝑝(𝐵𝑦, 𝑇𝑦).

(ii) Φ ⊂ Ψ. Browder and Petryshyn [30] introduced the notion of asymptotic regularity for a single valued mapping in a metric space (see also [3], page 547). Definition 17. A self-mapping 𝑇 of a metric space (𝑋, 𝑑) is asymptotically regular at a point 𝑥 ∈ 𝑋 if (14)

If 𝑇 is asymptotically regular at each point of 𝑋 then one says that 𝑇 is asymptotically regular on 𝑋. Sastry et al. [31] and Singh et al. [32] extended the above definition to three mappings as follows. Definition 18. Let 𝑆, 𝑇, and 𝑓 be self-mappings of a metric space (𝑋, 𝑑). The pair (𝑆, 𝑇) is asymptotically regular with respect to 𝑓 at a point 𝑥0 ∈ 𝑋 if there exists a sequence {𝑥𝑛 } in 𝑋 such that (15)

(18)

(19)

(i) 𝐴 and 𝑆 have a coincidence point; (ii) 𝐵 and 𝑇 have a coincidence point. Moreover, if the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are weakly compatible, then 𝐴, 𝐵, 𝑆, and 𝑇 have a unique common fixed point. Now we present a more general result than Theorem 20. Theorem 21. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of a partial metric space (𝑋, 𝑝) such that (A) 𝐴𝑋 ⊆ 𝑇𝑋 and 𝐵𝑋 ⊆ 𝑆𝑋; (B) 𝐴, 𝐵, 𝑆, and 𝑇 are asymptotically regular at 𝑥0 ∈ 𝑋; 𝑝 (𝐴𝑥, 𝐵𝑦) ≤ 𝜓 (𝛼 (𝑥, 𝑦)) ,

(20)

for all 𝑥, 𝑦 ∈ 𝑋, where 𝜓 ∈ Ψ. If one of 𝐴𝑋, 𝐵𝑋, 𝑇𝑋, and 𝑆𝑋 is a 0-complete subspace of 𝑋, then (a) 𝐴 and 𝑆 have a coincidence point; (b) 𝐵 and 𝑇 have a coincidence point;

for all 𝑛 ∈ N ∪ {0} and 𝑛→∞

lim 𝑝 (𝑦𝑛 , 𝑦𝑛+1 ) = 0.

𝑛→∞

for all 𝑥, 𝑦 ∈ 𝑋, where 𝜙 ∈ Φ. If one of 𝐴𝑋, 𝐵𝑋, 𝑇𝑋, and 𝑆𝑋 is a closed subset of (𝑋, 𝑝), then

(i) 𝑀(𝑥, 𝑦) ≤ 𝛼(𝑥, 𝑦) for all 𝑥, 𝑦 ∈ 𝑋 (see [8]);

lim 𝑑 (𝑓𝑥𝑛 , 𝑓𝑥𝑛+1 ) = 0.

𝑛 ∈ N ∪ {0} and

𝑝 (𝐴𝑥, 𝐵𝑦) ≤ 𝜙 (𝑀 (𝑥, 𝑦)) ,

Remark 16. It can be easily seen that

𝑓𝑥2𝑛+2 = 𝑇𝑥2𝑛+1

(17)

Theorem 20. Suppose 𝐴, 𝐵, 𝑆, and 𝑇 are self-mappings of a complete partial metric space (𝑋, 𝑝) such that 𝐴𝑋 ⊆ 𝑇𝑋, 𝐵𝑋 ⊆ 𝑆𝑋, and

(3) 𝑀(𝑥, 𝑦) = max{𝑝(𝑆𝑥, 𝑇𝑦), 𝑝(𝐴𝑥, 𝑆𝑥), 𝑝(𝐵𝑦, 𝑇𝑦), (1/2)[𝑝(𝑆𝑥, 𝐵𝑦) + 𝑝(𝐴𝑥, 𝑇𝑦)]};

𝑓𝑥2𝑛+1 = 𝑆𝑥2𝑛 ,

𝑦2𝑛+1 = 𝑆𝑥2𝑛+2 = 𝐵𝑥2𝑛+1 .

The following theorem is the main result in [10].

Let 𝐴, 𝐵, 𝑆, 𝑇 : 𝑋 → 𝑋 be mappings.

lim 𝑑 (𝑇𝑛 𝑥, 𝑇𝑛+1 𝑥) = 0.

𝑦2𝑛 = 𝑇𝑥2𝑛+1 = 𝐴𝑥2𝑛 ;

+

(1) Φ the class of functions 𝜙 : R → R such that 𝜙 is continuous nondeceasing function satisfying 𝜙(𝑡) < 𝑡 𝑛 and the series ∑∞ 𝑛≥1 𝜙 (𝑡) converges for all 𝑡 > 0;

𝑛→∞

asymptotically regular at 𝑥0 ∈ 𝑋 if there exist sequences {𝑥𝑛 } and {𝑦𝑛 } in 𝑋 such that

(16)

If 𝑆 = 𝑇 then one gets the definition of asymptotic regularity of 𝑇 with respect to 𝑓 (see, for instance, Rhoades et al. [33]). Further, if 𝑓 is the identity mapping on 𝑋, then one gets the usual definition of asymptotic regularity for a mapping 𝑇. We extend the above notion to four self-mappings on a partial metric space as follows. Definition 19. Let 𝐴, 𝐵, 𝑆, and 𝑇 be self-mappings of a partial metric space (𝑋, 𝑝). The mappings 𝐴, 𝐵, 𝑆, and 𝑇 will be called

(c) 𝐴 and 𝑆 have a common fixed point provided that the pair (𝐴, 𝑆) is commuting at one of their coincidence points; (d) 𝐵 and 𝑇 have a common fixed point provided that the pair (𝐵, 𝑇) is commuting at one of their coincidence points. Moreover, the mappings 𝐴, 𝐵, 𝑆, and 𝑇 have a common fixed point provided that (𝑐) and (𝑑) are true. Proof. Let 𝑥0 ∈ 𝑋 be such that 𝐴, 𝐵, 𝑆, and 𝑇 are asymptotically regular at 𝑥0 . Since 𝐴𝑋 ⊆ 𝑇𝑋, there exists 𝑥1 ∈ 𝑋 such that 𝑇𝑥1 = 𝐴𝑥0 . Also since 𝐵𝑋 ⊆ 𝑆𝑋, there exists 𝑥2 ∈ 𝑋


5

such that 𝑆𝑥2 = 𝐵𝑥1 . Continuing this process, we construct sequences {𝑥𝑛 } and {𝑦𝑛 } in 𝑋 defined by 𝑦2𝑛 = 𝑇𝑥2𝑛+1 = 𝐴𝑥2𝑛 ; 𝑦2𝑛+1 = 𝑆𝑥2𝑛+2 = 𝐵𝑥2𝑛+1 ,

(21)

where 𝑛 ∈ N ∪ {0}. Since 𝐴, 𝐵, 𝑆, and 𝑇 are asymptotically regular at 𝑥0 , we have lim 𝑝 (𝑦𝑛 , 𝑦𝑛+1 ) = 0.

𝑛→∞

(22)

We claim that {𝑦𝑛 } is a 0-Cauchy sequence. Suppose {𝑦𝑛 } is not 0-Cauchy. Then there exist 𝛽 > 0 and increasing sequences {𝑚𝑘 } and {𝑛𝑘 } of positive integers such that 𝑝 (𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 +1 ) ≥ 𝛽, 𝑝 (𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) < 𝛽,

(23)

for all 𝑛 ≤ 2𝑚𝑘 < 2𝑛𝑘 + 1. By the triangle inequality, we have 𝑝 (𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 +1 ) ≤ 𝑝 (𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) + 𝑝 (𝑦2𝑛𝑘 , 𝑦2𝑛𝑘 +1 ) − 𝑝 (𝑦2𝑛𝑘 , 𝑦2𝑛𝑘 )

(24)

Thus 𝑝(𝐴𝑢, 𝐵V) ≤ 𝜓(𝑝(𝐴𝑢, 𝐵V)) < 𝑝(𝐴𝑢, 𝐵V), a contradiction, unless 𝑝(𝐴𝑢, 𝐵V) = 0. Therefore 𝑇V = 𝐴𝑢 = 𝐵V and V is a coincidence point of 𝐵 and 𝑇. If the pairs (𝐴, 𝑆) and (𝐵, 𝑇) are commuting at 𝑢 and V, respectively, then 𝐴𝑆𝑢 = 𝑆𝐴𝑢, 𝐴𝐴𝑢 = 𝐴𝑆𝑢 = 𝑆𝐴𝑢 = 𝑆𝑆𝑢;

(29)

𝑇𝑇V = 𝑇𝐵V = 𝐵𝑇V = 𝐵𝐵V.

(30)

= 𝜓 (𝑝 (𝐴𝐴𝑢, 𝐴𝑢)) , (25)

= 𝜓 (𝑝 (𝑦2𝑚𝑘 −1 , 𝑦2𝑛𝑘 ) + 𝑝 (𝑦2𝑚𝑘 , 𝑦2𝑚𝑘 −1 ) + 𝑝 (𝑦2𝑛𝑘 +1 , 𝑦2𝑛𝑘 )) . Since 𝜓 is upper semicontinuous from the right, we deduce that

a contradiction. Therefore 𝐴𝐴𝑢 = 𝐴𝑢 = 𝑆𝐴𝑢 and 𝐴𝑢 is a common fixed point of 𝐴 and 𝑆. Similarly, 𝐵V is a common fixed point of 𝐵 and 𝑇. Since 𝐴𝑢 = 𝐵V, we conclude that 𝐴𝑢 is a common fixed point of 𝐴, 𝐵, 𝑆, and 𝑇. The proof is similar when 𝑇𝑋 is a complete subspace of 𝑋. The cases in which 𝐴𝑋 or 𝐵𝑋 is a complete subspace of 𝑋 are also similar since 𝐴𝑋 ⊆ 𝑇𝑋 and 𝐵𝑋 ⊆ 𝑆𝑋.

(26)

a contradiction. Therefore lim𝑚,𝑛→∞ 𝑝(𝑦𝑛 , 𝑦𝑚 ) = 0. Suppose that 𝑆𝑋 is a 0-complete subspace of 𝑋. Then the subsequence {𝑦2𝑛 } being contained in 𝑆𝑋 has a limit in 𝑆𝑋. Call it 𝑧. Let 𝑢 ∈ 𝑆−1 𝑧. Then 𝑆𝑢 = 𝑧. Note that the subsequence {𝑦2𝑛+1 } also converges to 𝑧. By (20), we have 𝑝 (𝐴𝑢, 𝐵𝑥2𝑛+1 ) ≤ 𝜓 (𝛼 (𝑢, 𝑥2𝑛+1 ))

+ 𝑝 (𝐵𝑥2𝑛+1 , 𝑇𝑥2𝑛+1 )) = 𝜓 (𝑝 (𝑆𝑢, 𝑦2𝑛 )

= 𝜓 (𝑝 (𝐵V, 𝐴𝑢)) .

= 𝜓 (𝑝 (𝑆𝐴𝑢, 𝑇V) + 𝑝 (𝐴𝐴𝑢, 𝑆𝐴𝑢) + 𝑝 (𝐵V, 𝑇V))

≤ 𝜓 (𝛼 (𝑥2𝑚𝑘 , 𝑥2𝑛𝑘 +1 )) = 𝜓 (𝑝 (𝑆𝑥2𝑚𝑘 , 𝑇𝑥2𝑛𝑘 +1 )

= 𝜓 (𝑝 (𝑆𝑢, 𝑇𝑥2𝑛+1 ) + 𝑝 (𝐴𝑢, 𝑆𝑢)

(28)

𝑝 (𝐴𝐴𝑢, 𝐴𝑢) = 𝑝 (𝐴𝐴𝑢, 𝐵V) ≤ 𝜓 (𝛼 (𝐴𝑢, V))

𝑝 (𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 +1 ) = 𝑝 (𝐴𝑥2𝑚𝑘 , 𝐵𝑥2𝑛𝑘 +1 )

𝑘→∞

= 𝜓 (𝑝 (𝑆𝑢, 𝑇V) + 𝑝 (𝐴𝑢, 𝑆𝑢) + 𝑝 (𝐵V, 𝑇V))

Now, in view of (20), it follows that

Thus lim𝑘→∞ 𝑝(𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 +1 ) = 𝛽. Now, by (20), we have

𝛽 ≤ lim sup 𝜓 (𝛼 (𝑥2𝑚𝑘 , 𝑥2𝑛𝑘 +1 )) ≤ 𝜓 (𝛽) ,

𝑝 (𝐴𝑢, 𝐵V) ≤ 𝜓 (𝛼 (𝑢, V))

𝐵𝑇V = 𝑇𝐵V,

≤ 𝑝 (𝑦2𝑚𝑘 , 𝑦2𝑛𝑘 ) + 𝑝 (𝑦2𝑛𝑘 , 𝑦2𝑛𝑘 +1 ) .

+ 𝑝 (𝐴𝑥2𝑚𝑘 , 𝑆𝑥2𝑚𝑘 ) + 𝑝 (𝐵𝑥2𝑛𝑘 +1 , 𝑇𝑥2𝑛𝑘 +1 ))

contradiction, unless 𝑝(𝐴𝑢, 𝑆𝑢) = 0. Therefore 𝐴𝑢 = 𝑆𝑢 = 𝑧 and 𝑢 is a coincidence point of 𝐴 and 𝑆. Since 𝐴𝑋 ⊆ 𝑇𝑋, 𝑧 = 𝐴𝑢 ∈ 𝑇𝑋. Hence there exists V ∈ 𝑋 such that 𝐴𝑢 = 𝑇V. Again, by (20), we have

When 𝐴 = 𝐵 and 𝑆 = 𝑇 = id (the identity mapping) in Theorem 21, we get the following result which extends a result of Romaguera [23]. Corollary 22. Let 𝐴 be an asymptotically regular self-mapping of a partial metric space (𝑋, 𝑝) such that 𝑝 (𝐴𝑥, 𝐴𝑦) ≤ 𝜓 (𝜇𝑝 (𝑥, 𝑦)) ,

(27)

+ 𝑝 (𝐴𝑢, 𝑆𝑢) + 𝑝 (𝑦2𝑛+1 , 𝑦2𝑛 )) . Since 𝜓 is upper semicontinuous from the right, making 𝑘 → ∞ implies 𝑝(𝐴𝑢, 𝑆𝑢) ≤ 𝜓(𝑝(𝐴𝑢, 𝑆𝑢)) < 𝑝(𝐴𝑢, 𝑆𝑢), a

(31)

for all 𝑥, 𝑦 ∈ 𝑋, where 𝜓 ∈ Ψ. Then 𝐴 has a fixed point. The following example shows the generality of our results. Example 23. Let 𝑋 = {0, 1, 2} endowed with the partial metric 𝑝(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ 𝑋. Then (𝑋, 𝑝)

6


is a 0-complete partial metric space. Define the mappings 𝐴, 𝐵, 𝑆, 𝑇 : 𝑋 → 𝑋 by

for all 𝑥, 𝑦 ∈ 𝑋, where 𝛼(𝑥, 𝑦) = 𝑝(𝑆𝑥, 𝑇𝑦) + 𝑝(𝐴𝑥, 𝑆𝑥) + 𝑝(𝐵𝑦, 𝑆𝑦). For this, let 𝑥, 𝑦 ∈ 𝑋 with 𝑦 ≤ 𝑥. Then 1 1 1 𝑝 (𝐴𝑥, 𝐵𝑦) = 𝑝 ( 𝑥2 , 𝑦2 ) = 𝑥2 , 2 8 2

𝐴 = 𝐵, 𝑆 = 𝑇,

𝛼 (𝑥, 𝑦) = 𝑝 (𝑆𝑥, 𝑇𝑦) + 𝑝 (𝐴𝑥, 𝑆𝑥) + 𝑝 (𝐵𝑦, 𝑆𝑦)

𝐴0 = 𝐴1 = 0, 𝐴2 = 2;

1 1 1 = 𝑝 ( 𝑥2 , 𝑦2 ) + 𝑝 ( 𝑥2 , 𝑥2 ) 6 2 6

(32)

1 1 1 2 1 + 𝑝 ( 𝑦2 , 𝑦2 ) ≥ 𝑥2 + 𝑥2 = 𝑥2 . 8 6 6 2 3

𝑇0 = 0, 𝑇1 = 1,

Then

𝑇2 = 2.

2 1 𝜓 (𝛼 (𝑥, 𝑦)) ≥ 𝜓 ( 𝑥2 ) = 𝑥2 = 𝑝 (𝐴𝑥, 𝐵𝑦) . 3 2

Define sequences {𝑥𝑛 } and {𝑦𝑛 } by

(33)

2

𝑦2𝑛 = 𝑇𝑥2𝑛+1 = 𝐴𝑥2𝑛 = (𝑥2𝑛+1 ) =

𝑦2𝑛+1 = 𝑆𝑥2𝑛+2 = 𝐵𝑥2𝑛+1 , where 𝑛 ∈ N ∪ {0}. Then the mappings 𝐴, 𝐵, 𝑆, and 𝑇 are asymptotically regular at 0. Further, 𝑝 (𝐴𝑥, 𝐵𝑦) ≤ 𝜓 (𝛼 (𝑥, 𝑦)) ,

(38)

Now we show that the mappings 𝐴, 𝐵, 𝑆, and 𝑇 are asymptotically regular. For this, by Definition 19, we have for all 𝑛 ∈ N ∪ {0}

𝑥0 = 𝑥𝑛 = 0; 𝑦2𝑛 = 𝑇𝑥2𝑛+1 = 𝐴𝑥2𝑛 ;

(37)

(34)

for all 𝑥, 𝑦 ∈ 𝑋, where 𝛼(𝑥, 𝑦) = 𝑝(𝑆𝑥, 𝑇𝑦) + 𝑝(𝐴𝑥, 𝑆𝑥) + 𝑝(𝐵𝑦, 𝑆𝑦) and 𝜓(𝑡) = 3𝑡/4. So the assumptions of Theorem 21 are fulfilled and 𝐴, 𝐵, 𝑆, and 𝑇 have fixed points 0 and 2. On the other hand, 𝑝(𝐴1, 𝐵2) > 𝜙(𝑀(1, 2)) for any 𝜙. Therefore the mappings 𝐴, 𝐵, 𝑆, and 𝑇 do not satisfy the requirements of Theorem 20. Example 24. Let 𝑋 = [0, 1] be endowed with the partial metric 𝑝(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ 𝑋. Then (𝑋, 𝑝) is a 0-complete partial metric space. Define the mappings 𝐴, 𝐵, 𝑆, 𝑇 : 𝑋 → 𝑋 by

𝑦2𝑛+1 = 𝑆𝑥2𝑛+2 = 𝐵𝑥2𝑛+1

1 2 (𝑥 ) , 2 2𝑛

1 1 2 2 = (𝑥2𝑛+2 ) = (𝑥2𝑛+1 ) . 6 8

(39)

Solving the above two equations we get 𝑥2𝑛+1 = 𝑥2𝑛+2

1 𝑥 , √2 2𝑛

3 = √ 𝑥2𝑛 . 8

(40)

So for given 𝑥0 ∈ 𝑋 we can define a sequence {𝑥𝑛 } by using (40) and then we can easily define sequence {𝑦𝑛 } by (39). It can be easily seen that lim𝑛→∞ 𝑝(𝑦𝑛 , 𝑦𝑛+1 ) = 0. On the other hand, for 𝑦 = 0 and 𝑥 > 0, we have 1 1 𝑝 (𝐴𝑥, 𝐵0) = 𝑝 ( 𝑥2 , 0) = 𝑥2 , 2 2 𝑀 (𝑥, 𝑦) = max {𝑝 (𝑆𝑥, 𝑇0) , 𝑝 (𝐴𝑥, 𝑆𝑥) , 𝑝 (𝐵0, 𝑇0) ,

1 𝐴𝑥 = 𝑥2 , 2

1 1 [𝑝 (𝑆𝑥, 𝐵0) + 𝑝 (𝐴𝑥, 𝑇0)]} = max {𝑝 ( 𝑥2 , 0) , (41) 2 6

1 𝐵𝑥 = 𝑥2 , 8

1 1 𝑝 ( 𝑥2 , 𝑥2 ) , 𝑝 (0, 0) , 2 6

1 𝑆𝑥 = 𝑥2 , 6

(35)

1 1 1 1 [𝑝 ( 𝑥2 , 0) + 𝑝 ( 𝑥2 , 0)]} = 𝑥2 . 2 6 2 2

𝑇𝑥 = 𝑥2 . Then 𝐴𝑋 ⊆ 𝑇𝑋 and 𝐵𝑋 ⊆ 𝑆𝑋. Now define the function 𝜓 : R+ → R+ by 𝜓(𝑡) = (3/4)𝑡. We now show that 𝑝 (𝐴𝑥, 𝐵𝑦) ≤ 𝜓 (𝛼 (𝑥, 𝑦)) ,

(36)

Therefore 𝑝(𝐴𝑥, 𝐵0) = (1/2)𝑥2 = 𝑀(𝑥, 𝑦) > 𝜙(𝑀(𝑥, 𝑦)) for any 𝜙. Therefore the mappings 𝐴, 𝐵, 𝑆, and 𝑇 do not satisfy the requirement of Theorem 20. Now we give an example in which the mapping is not asymptotically regular at each point of interval but satisfies condition (4) for one mapping.


7

Example 25. Let 𝑋 = [0, 1] be endowed with the partial metric 𝑝(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ 𝑋. Then (𝑋, 𝑝) is a 0-complete partial metric space. Define the mapping by 1 { {0, if 𝑥 ∈ [0, 2 ] , 𝑇𝑥 = { {1, otherwise 𝑥 ∈ ( 1 , 1] . { 2

(42)

Now define the function 𝜑 : R+ → R+ by 𝜑(𝑡) = (3/4)𝑡. It can be easily seen that 𝑇 is asymptotically regular at each point of interval [0, 1/2] and not asymptotically regular at any point of interval (1/2, 1]. Now we show that 𝑝 (𝐴𝑥, 𝐵𝑦) ≤ 𝜑 (𝛼 (𝑥, 𝑦)) ,

(43)

for all 𝑥, 𝑦 ∈ 𝑋, where 𝛼(𝑥, 𝑦) = 𝑝(𝑥, 𝑦)+𝑝(𝑥, 𝑇𝑥)+𝑝(𝑦, 𝑇𝑦). For this we distinguish the following cases. Case 1. If 𝑥, 𝑦 ∈ [0, 1/2] with 𝑥 ≤ 𝑦, we have 𝑝 (𝑇𝑥, 𝑇𝑦) = 0 ≤ 𝜑 (𝛼 (𝑥, 𝑦)) .

(44)

Case 2. If 𝑥 ∈ [0, 1/2] and 𝑦 ∈ (1/2, 1], we have 3 1 9 3 (𝑦 + 𝑥 + 1) ≥ ( + 1) = > 1 4 4 2 8

(45)

𝜑 (𝛼 (𝑥, 𝑦)) = 𝜑 (𝑝 (𝑥, 𝑦) + 𝑝 (𝑥, 𝑇𝑥) + 𝑝 (𝑦, 𝑇𝑦)) (46)

𝜇 (𝑥, 𝑦) = 𝑝 (𝑥, 𝑦) + 𝛾 [𝑝 (𝑥, 𝑆𝑥) + 𝑝 (𝑦, 𝑆𝑦)] .

(49)

If 𝑆 is asymptotically regular at 𝑥0 ∈ 𝑋, then 𝑆 has a fixed point. Moreover, if 𝛾 = 1 and 𝜑 is continuous and satisfies 𝜑(𝑡) < 𝑡 for all 𝑡 > 0, then the continuity of 𝑆 can be dropped.

4. Multivalued Mappings Rhoades et al. [33] and Singh et al. [32] extended the concept of asymptotic regularity from single valued to multivalued mappings in metric spaces. We extend it to partial metric spaces. Definition 26. Let (𝑋, 𝑝) be a partial metric space and 𝑆 : 𝑋 → 𝐶𝐵𝑝 (𝑋). The mapping 𝑆 is asymptotically regular at 𝑥0 ∈ 𝑋 if, for any sequence {𝑥𝑛 } in 𝑋 and each sequence {𝑦𝑛 } in 𝑋 such that 𝑦𝑛 ∈ 𝑆𝑥𝑛−1 , 𝑛→∞

(S1) 𝐻𝑝 (𝑆𝑥, 𝑆𝑦) ≤ 𝜑(𝜇(𝑥, 𝑦)) for all 𝑥, 𝑦 ∈ 𝑋; where 𝜑 is as in Theorem 3, 𝐶𝑝 (𝑋) is a collection of all nonempty compact subsets of 𝑋, 𝛾 ≥ 0, and

Case 3. If 𝑥, 𝑦 ∈ (1/2, 1] with 𝑥 ≤ 𝑦, we have

lim 𝑝 (𝑦𝑛 , 𝑦𝑛+1 ) = 0

A self-mapping 𝑇 : 𝑋 → 𝑋 and a multivalued mapping 𝑇 : 𝑋 → 𝐶𝐵𝑝 (𝑋) both are mixed multivalued mappings (see also [35]). Motivated by Proinov’s theorem and the above facts, we obtain the following result, which extends Theorem 27 above and Corollary 2.5 in [16].

(S2) 𝐻𝑝 (𝑆𝑥, 𝑆𝑦) < 𝜇(𝑥, 𝑦) whenever 𝜇(𝑥, 𝑦) ≠ 0,

= 𝑝 (𝑇𝑥, 𝑇𝑦) .

3 3 = (𝑦 + 1 + 1) ≥ > 1 = 𝑝 (𝑇𝑥, 𝑇𝑦) . 4 2

Definition 28. Let (𝑋, 𝑝) be a partial metric space. A mapping 𝑇 : 𝑋 → 𝑋 ∪ 𝐶𝐵𝑝 (𝑋) is called a mixed multivalued mapping on 𝑋 if 𝑇 is a multivalued mapping on 𝑋 such that for each 𝑥 ∈ 𝑋 either 𝑇𝑥 ∈ 𝑋 or 𝑇𝑥 ∈ 𝐶𝐵𝑝 (𝑋).

Theorem 29. Let (𝑋, 𝑝) be a 0-complete partial metric space and 𝑆 : 𝑋 → 𝑋 ∪ 𝐶𝑝 (𝑋) a continuous mixed multivalued mapping such that

𝜑 (𝛼 (𝑥, 𝑦)) = 𝜑 (𝑝 (𝑥, 𝑦) + 𝑝 (𝑥, 𝑇𝑥) + 𝑝 (𝑦, 𝑇𝑦)) =

In [24], Romaguera pointed out that if 𝑋 = R+ and 𝑝 : 𝑋 × 𝑋 → R+ defined by 𝑝(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ 𝑋, then 𝐶𝐵𝑝 (𝑋) = 0 and the approach used in Theorem 27 and elsewhere has a disadvantage that the fixed point theorems for self-mappings may not be derived from it, when 𝐶𝐵𝑝 (𝑋) = 0. To overcome this problem he introduced the concept of mixed multivalued mappings and obtained a different version of Nadler Jr.’s theorem in a partial metric space.

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for 𝑛 ∈ N ∪ {0}.

Proof. We construct a sequence {𝑥𝑛 } in 𝑋 in the following way. Let 𝑥0 ∈ 𝑋 such that 𝑆 is asymptotically regular at 𝑥0 . Let 𝑥1 be any element of 𝑆𝑥0 . If 𝑥0 = 𝑥1 or 𝑥1 ∈ 𝑆𝑥1 , then 𝑥1 is a fixed point of 𝑆 and there is nothing to prove. Assume that 𝑥1 ∉ 𝑆𝑥1 and 𝑆𝑥1 is not singleton. Then 𝑆𝑥1 ∈ 𝐶𝑝 (𝑋) and by compactness of 𝑆𝑥1 we can choose 𝑥2 ∈ 𝑆𝑥1 such that 𝑝 (𝑥1 , 𝑥2 ) ≤ 𝐻𝑝 (𝑆𝑥0 , 𝑆𝑥1 ) .

(50)

If 𝑆𝑥1 = {𝑥2 } is a singleton, then obviously 𝑝 (𝑥1 , 𝑥2 ) ≤ 𝐻𝑝 (𝑆𝑥0 , 𝑆𝑥1 ) .

(51)

Therefore, in either case, we have

Aydi et al. [14] obtained the following equivalent to the well-known multivalued contraction theorem due to Nadler Jr. [34]. Theorem 27. Let (𝑋, 𝑝) be a complete partial metric space. If 𝑇 : 𝑋 → 𝐶𝐵𝑝 (𝑋) is a multivalued mapping such that, for all 𝑥, 𝑦 ∈ 𝑋 and 𝑘 ∈ (0, 1), one has 𝐻𝑝 (𝑇𝑥, 𝑇𝑦) ≤ 𝑘𝑝 (𝑥, 𝑦) , then 𝑇 has a fixed point.

(48)

𝑝 (𝑥1 , 𝑥2 ) ≤ 𝐻𝑝 (𝑆𝑥0 , 𝑆𝑥1 ) .

(52)

Again, since 𝑆𝑥2 is compact, we choose a point 𝑥3 ∈ 𝑆𝑥2 such that 𝑝 (𝑥2 , 𝑥3 ) ≤ 𝐻𝑝 (𝑆𝑥1 , 𝑆𝑥2 ) .

(53)

Continuing in the same manner we get 𝑝 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝐻𝑝 (𝑆𝑥𝑛−1 , 𝑆𝑥𝑛 ) .

(54)

8


Following largely [3, 8], we show that the sequence {𝑥𝑛 } is a 0-Cauchy. Fix 𝜀 > 0. Since 𝜑 is as in Theorem 3, there exists 𝛿 > 𝜀 such that, for any 𝑡 ∈ (0, ∞), 𝜀 < 𝑡 < 𝛿 󳨐⇒

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𝜑 (𝑡) ≤ 𝜀.

Without loss of generality we may assume that 𝛿 ≤ 2𝜀. Since 𝑆 is asymptotically regular at 𝑥0 , lim 𝑝 (𝑥𝑛 , 𝑥𝑛+1 ) = 0.

(56)

𝑛→∞

So, there exists an integer 𝑁1 ≥ 1 such that 𝑝 (𝑥𝑛 , 𝑥𝑛+1 ) ≤ 𝐻𝑝 (𝑆𝑥𝑛−1 , 𝑆𝑥𝑛 )
0, then it follows from (S2) that

𝑝 (𝑥𝑛 , 𝑥𝑚+1 ) ≤ 𝑝 (𝑥𝑛 , 𝑥𝑛+1 ) + 𝑝 (𝑥𝑛+1 , 𝑥𝑚+1 )

≤ 𝑝 (𝑥𝑛 , 𝑥𝑛+1 ) + 𝑝 (𝑥𝑛+1 , 𝑥𝑚+1 )

𝑘 ⋅ 8 4 40

𝑇𝑖 𝑢 (𝑡) = 𝑔 (𝑡) + ∫ 𝐾𝑖 (𝑡, 𝑠, 𝑢 (𝑠)) 𝑑𝑠 0

(89)

∀𝑢 ∈ 𝐶 (𝐼, R) , 𝑡 ∈ 𝐼, 𝑖 ∈ {1, 2, 3, 4} .

3 3 17 = max { , } = . 4 32 4

(85)

= 𝑘 ⋅ 𝑝 (𝑥, 𝑦)

Clearly, 𝑢 is a solution of (88) if and only if it is a common fixed point of 𝑇𝑖 for 𝑖 ∈ {1, 2, 3, 4}. We shall prove the existence of a common fixed point of 𝑇𝑖 for 𝑖 ∈ {1, 2, 3, 4} under certain conditions. Theorem 35. Suppose that the following hypotheses hold: (H1) For any 𝑢 ∈ 𝐶(𝐼, R), there exist 𝑘1 , 𝑘2 ∈ 𝐶(𝐼, R) such that 𝑇1 𝑢 = 𝑇3 𝑘1 , 𝑇2 𝑢 = 𝑇4 𝑘2 .

for all 𝑘 ∈ [0, 1) and

(H2) For all 𝑡 ∈ 𝐼, 𝑢 ∈ 𝐶(𝐼, R),

𝑀 (𝑥, 𝑦) = max {𝑝 (𝑥, 𝑦) , 𝑝 (𝑥, 𝑆𝑥) , 𝑝 (𝑦, 𝑆𝑦) , 1 23 17 23 177 [𝑝 (𝑥, 𝑆𝑦) + 𝑝 (𝑦, 𝑆𝑥)]} = max { , , , } (86) 2 40 32 40 320 =

This section is inspired by the work given in paper [37] and the purpose of this section is to give an existence theorem for a solution of (88) given below. Let 𝐼 = [0, 𝐾] ⊂ R be a closed and bounded interval with 𝐾 > 0. Consider the system of Volterra type integral equations:

23 . 40

𝑇1 𝑇4 𝑢 (𝑡) = 𝑇4 𝑇1 𝑢 (𝑡) ,

whenever 𝑇1 𝑢 (𝑡) = 𝑇4 𝑢 (𝑡) , (90)

and for all 𝑡 ∈ 𝐼, 𝑢 ∈ 𝐶(𝐼, R), 𝑇2 𝑇3 𝑢 (𝑡) = 𝑇3 𝑇2 𝑢 (𝑡) ,

Hence 𝐻𝑝 (𝑆𝑥, 𝑆𝑦) =

3 23 >𝑘⋅ = 𝑘 ⋅ 𝑀 (𝑥, 𝑦) 4 40

(87)

for all 𝑘 ∈ [0, 1). Remark 34. We remark that the mappings satisfying Theorems 2, 27, and 20 are also asymptotically regular. Therefore the extra assumption of asymptotical regularity in our results is not strong.

whenever 𝑇2 𝑢 (𝑡) = 𝑇3 𝑢 (𝑡) . (91)

(H3) There exists a continuous function ℏ : 𝐼 × 𝐼 → R+ such that for all 𝑡, 𝑠 ∈ 𝐼 and 𝑢, V ∈ 𝐶(𝐼, R) 󵄨 󵄨󵄨 󵄨󵄨𝐾1 (𝑡, 𝑠, 𝑢 (𝑠)) − 𝐾2 (𝑡, 𝑠, V (𝑠))󵄨󵄨󵄨 ≤ ℏ (𝑡, 𝑠) 󵄨 󵄨 󵄨 󵄨 ⋅ [󵄨󵄨󵄨𝑇4 𝑢 (𝑠) − 𝑇3 V (𝑠)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑇1 𝑢 (𝑠) − 𝑇4 𝑢 (𝑠)󵄨󵄨󵄨 󵄨 󵄨 + 󵄨󵄨󵄨𝑇2 V (𝑠) − 𝑇3 V (𝑠)󵄨󵄨󵄨] .

(92)


11

(H4) For sequences {ℎ𝑛 } and {𝑘𝑛 } in 𝐶(𝐼, R) such that 𝑘2𝑛 = 𝑇3 ℎ2𝑛 = 𝑇1 ℎ2𝑛 ; 𝑘2𝑛+1 = 𝑇4 ℎ𝑛 = 𝑇2 ℎ2𝑛+1 ,

𝑢, V ∈ 𝑋 such that ‖𝑢‖𝜏 , ‖V‖𝜏 ≤ 1, by assertions (H3)–(H5), we have (93)

𝑡

󵄨󵄨 󵄨 󵄨 󵄨󵄨𝑇1 𝑢 (𝑡) − 𝑇2 V (𝑡)󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨𝐾1 (𝑡, 𝑠, 𝑢 (𝑡)) 0 󵄨 − 𝐾2 (𝑡, 𝑠, V (𝑡))󵄨󵄨󵄨 𝑑𝑠

one has

𝑡

󵄨 󵄨 lim 󵄨󵄨𝑘 (𝑡) − 𝑘𝑛+1 (𝑡)󵄨󵄨󵄨 = 0 ∀𝑡 ∈ 𝐼, 𝑛 ∈ N ∪ {0} . 𝑛→∞ 󵄨 𝑛

(94)

≤ (∫

0

3𝜏 󵄨󵄨 󵄨 󵄨 󵄨 [󵄨𝑇 𝑢 (𝑠) − 𝑇3 V (𝑠)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑇1 𝑢 (𝑠) − 𝑇4 𝑢 (𝑠)󵄨󵄨󵄨 4 󵄨 4

󵄨 󵄨 + 󵄨󵄨󵄨𝑇2 V (𝑠) − 𝑇3 V (𝑠)󵄨󵄨󵄨] 𝑑𝑠)

𝑡

(H5) sup𝑡∈𝐼 ∫ ℏ(𝑡, 𝑠)𝑑𝑠 ≤ 3/4. 0

Then system (88) of integral equations has a solution 𝑢∗ ∈ 𝐶(𝐼, R). = Proof. For 𝑥 ∈ 𝑋 = 𝐶(𝐼, R) define ‖𝑥‖𝜏 max𝑡∈[0,𝐾] {|𝑥(𝑡)|𝑒−𝜏𝑡 }, where 𝜏 ≥ 1 is taken arbitrary. Notice that ‖ ⋅ ‖𝜏 is a norm equivalent to the maximum norm and (𝑋, ‖ ⋅ ‖𝜏 ) is a Banach space (cf. [38, 39]). The metric induced by this norm is given by 󵄨 󵄨 𝑑𝜏 (𝑥, 𝑦) = max {󵄨󵄨󵄨𝑥 (𝑡) − 𝑦 (𝑡)󵄨󵄨󵄨 𝑒−𝜏𝑡 } , 𝑡∈[0,𝐾]

(95)

for all 𝑥, 𝑦 ∈ 𝑋. Now, consider 𝑋 endowed with the partial metric given by 󵄩 󵄩 if ‖𝑥‖𝜏 , 󵄩󵄩󵄩𝑦󵄩󵄩󵄩𝜏 ≤ 1,

{𝑑𝜏 (𝑥, 𝑦) , 𝑝𝜏 (𝑥, 𝑦) = { {𝑑𝜏 (𝑥, 𝑦) + 𝜏, otherwise.

(96)

Obviously, (𝑋, 𝑝𝜏 ) is 0-complete but not complete. In fact, the associated metric 𝑝𝜏𝑠 (𝑥, 𝑦) = 2𝑝𝜏 (𝑥, 𝑦) − 𝑝𝜏 (𝑥, 𝑥) − 𝑝𝜏 (𝑦, 𝑦) given by 𝑝𝜏𝑠 (𝑥, 𝑦) 󵄩 󵄩 󵄩 󵄩 if (‖𝑥‖𝜏 , 󵄩󵄩󵄩𝑦󵄩󵄩󵄩𝜏 ≤ 1) or (‖𝑥‖𝜏 , 󵄩󵄩󵄩𝑦󵄩󵄩󵄩𝜏 > 1) , (97) {2𝑑𝜏 (𝑥, 𝑦) , ={ {2𝑑𝜏 (𝑥, 𝑦) + 𝜏, otherwise,

is not complete [37]. It is evident that 𝑥∗ ∈ 𝑋 is a solution of (88) if and only if 𝑥∗ is a common fixed point of 𝑇𝑖󸀠 𝑠. By condition (H1), it is clear that 𝑇1 (𝐶 (𝐼, R)) ⊂ 𝑇3 (𝐶 (𝐼, R)) , 𝑇2 (𝐶 (𝐼, R)) ⊂ 𝑇4 (𝐶 (𝐼, R)) .

(98)

From condition (H2), the pairs (𝑇1 , 𝑇4 ) and (𝑇2 , 𝑇3 ) are commuting. Using condition (H3) and maximum norm, the mappings 𝑇𝑖 (𝑖 = 1, 2, 3, 4) are asymptotically regular on (𝑋, 𝑝𝜏 ). Now, we show that condition (20) holds. Observe that this condition needs not be checked for 𝑢 = V ∈ 𝑋. Then for

= (∫

𝑡

0

3𝜏 󵄨󵄨 󵄨 󵄨 󵄨 [󵄨𝑇 𝑢 (s) − 𝑇3 V (𝑠)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑇1 𝑢 (𝑠) − 𝑇4 𝑢 (𝑠)󵄨󵄨󵄨 4 󵄨 4

𝑡 3𝜏 󵄨 󵄨 + 󵄨󵄨󵄨𝑇2 V (𝑠) − 𝑇3 V (𝑠)󵄨󵄨󵄨] 𝑒−𝜏𝑠 𝑒𝜏𝑠 𝑑𝑠) = (∫ 𝑒𝜏𝑠 𝑑𝑠) 4 0 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ⋅ [󵄨󵄨𝑇4 𝑢 (𝑠) − 𝑇3 V (𝑠)󵄨󵄨 + 󵄨󵄨𝑇1 𝑢 (𝑠) − 𝑇4 𝑢 (𝑠)󵄨󵄨

(99)

3𝜏 𝑒𝜏𝑡 1 󵄨 󵄨 + 󵄨󵄨󵄨𝑇2 V (𝑠) − 𝑇3 V (𝑠)󵄨󵄨󵄨] 𝑒−𝜏𝑠 = ( − ) 4 𝜏 𝜏 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ⋅ [󵄩󵄩𝑇4 𝑢 (𝑠) − 𝑇3 V (𝑠)󵄩󵄩𝜏 + 󵄩󵄩𝑇1 𝑢 (𝑠) − 𝑇4 𝑢 (𝑠)󵄩󵄩󵄩𝜏 3𝜏 𝑒𝜏𝑡 󵄩󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑇2 V (𝑠) − 𝑇3 V (𝑠)󵄩󵄩󵄩𝜏 ] = ( ) [󵄩󵄩𝑇4 𝑢 (𝑠) 4 𝜏 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 − 𝑇3 V (𝑠)󵄩󵄩𝜏 + 󵄩󵄩𝑇1 𝑢 (𝑠) − 𝑇4 𝑢 (𝑠)󵄩󵄩𝜏 + 󵄩󵄩𝑇2 V (𝑠) 󵄩 − 𝑇3 V (𝑠)󵄩󵄩󵄩𝜏 ] .

Since ‖𝑢‖𝜏 , ‖V‖𝜏 ≤ 1, it follows that, for all 𝑢, V ∈ 𝑋 at least for some 𝑡 ∈ [0, 𝐾], we have 󵄨󵄨 󵄨 −𝜏𝑡 3 󵄩 󵄩 󵄨󵄨𝑇1 𝑥 (𝑡) − 𝑇2 𝑦 (𝑡)󵄨󵄨󵄨 𝑒 ≤ × [󵄩󵄩󵄩𝑇4 𝑢 (𝑠) − 𝑇3 V (𝑠)󵄩󵄩󵄩𝜏 4 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩𝑇1 𝑢 (𝑠) − 𝑇4 𝑢 (𝑠)󵄩󵄩󵄩𝜏 + 󵄩󵄩󵄩𝑇2 V (𝑠) − 𝑇3 V (𝑠)󵄩󵄩󵄩𝜏 ] .

(100)

Now, by considering the control functions 𝜓 : [0, +∞) → [0, +∞) defined by 𝜓(𝑡) = 3𝑡/4, for all 𝑡 > 0, we get 𝑝𝜏 (𝑇1 𝑢, 𝑇2 V) ≤ 𝜓 (𝑝𝜏 (𝑇4 𝑢 (𝑠) , 𝑇3 V (𝑠)) + 𝑝𝜏 (𝑇1 𝑢 (𝑠) , 𝑇4 𝑢 (𝑠)) + 𝑝𝜏 (𝑇2 V (𝑠) , 𝑇3 V (𝑠))) .

(101)

Putting 𝐴 = 𝑇1 , 𝐵 = 𝑇2 , 𝑇 = 𝑇3 , and 𝑆 = 𝑇4 , then all the hypotheses of Theorem 21 are satisfied. Therefore 𝐴, 𝐵, 𝑆, and 𝑇 have a common fixed point 𝑢∗ ∈ 𝐶(𝐼, R); that is, 𝑢∗ is a solution of system (88).

6. Conclusions The authors are able to present some general fixed point results for a wider class of mappings in partial metric spaces with illustrative examples and an application. Results presented herein cannot be directly obtained from the corresponding metric space versions.

12

Conflicts of Interest The authors declare that they do not have any conflicts of interest.

Authors’ Contributions First and second authors wrote results. Third author prepared application and fourth author worked out some examples. In this way, each author equally contributed to this paper and read and approved the final manuscript.

Acknowledgments The authors acknowledge the support of their institute/ university for providing basic research facilities.

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